Finite Group Databasebeta

← #35224366

Subgroups — #35224366

Order 20864.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782295C211yesmaximal, characteristic, complemented
2#23782137C211yesmaximal, characteristic
3#23781967C211yesmaximal, characteristic, complemented
4#23781964C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781956C211yesmaximal, characteristic, complemented
7#23781953C211yesmaximal, characteristic, complemented
8#23781945C211yesmaximal, characteristic, complemented
9#23781944C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#23781700C211yesmaximal, characteristic, complemented
12#23781690C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic
14#102×D163C211yesmaximal, characteristic
15#478×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579682C2211yescharacteristic
18C2×C8×D163C2211yescharacteristic
19#19104735C2211yescharacteristic, complemented
20#19104734—21no—
21#19104734C2211yescharacteristic, complemented
22#19104726C2211yescharacteristic, complemented
23#19104725—21no—
24#19104715—21no—
25#19104713—21no—
26C4×D652C2211yescharacteristic
27D4×Dic163C2211yescharacteristic
28#19104697C2211yescharacteristic
29C4×C22⋊D163—21no—
30(C22⋊D163)⋊C4—21no—
31#19104693C2211yescharacteristic
32#19104693C2211yescharacteristic
33#19104692C2211yescharacteristic
34C4⋊C4×D163C2211yescharacteristic
35C2×(C2×D163)⋊C4—21no—
36C22⋊C4×D163—21no—
37#19104668C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41#19104667C2211yescharacteristic, complemented
42#19104666C2211yescharacteristic, complemented
43C4⋊(C163⋊C8)C2211yescharacteristic, complemented
44Dic163⋊C8C2211yescharacteristic, complemented
45Dic163⋊C8C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50#19104655C2211yescharacteristic
51#19104655C2211yescharacteristic
52#125×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64D4×D163—22no—
65D4×D163Q814yescomplemented
66C22×Dic163—21no—
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163C2311yescharacteristic
71#14579682—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—22no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84Dic163⋊C4D411yescharacteristic
85C22⋊Dic163—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4D411yescharacteristic
91(C2×D163)⋊C4D411yescharacteristic
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×D4×C163C2311yescharacteristic
96C22×C4×C163—21no—
97C2×C8×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C42×C163C2311yescharacteristic
106C22×D163—21no—
107C22×D163—24no—
108C22×D163—81no—
109C22×D163—81no—
110C22×D163C2×D411yescharacteristic
111Dic326—81no—
112Dic326—81no—
113D652—21no—
114D652C2×Q812yes—
115C22⋊D163—24no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—81no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—81no—
124C4×D163—81no—
125C4×D163—81no—
126C4×D163C2×Q812yes—
127C4×D163#4112yes—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163—21no—
131D4×C163—21no—
132D4×C163C2×Q812yes—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—81no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2411yescharacteristic
140C8×C163—21no—
141C8×C163—21no—
142C22×C163—22no—
143C22×C163—41no—
144C22×C163—41no—
145C22×C163C22×D411yescharacteristic
146C2×D163—21no—
147C2×D163—21no—
148C2×D163—22no—
149C2×D163—41no—
150C2×D163—41no—
151C2×D163—84no—
152C2×D163C22⋊Q812yes—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—41no—
156C4×C163—41no—
157C4×C163—41no—
158C4×C163C22×Q811yescharacteristic
159C4×C163C2×#4111yescharacteristic
160Dic163—21no—
161Dic163—21no—
162Dic163—41no—
163Dic163—41no—
164Dic163—41no—
165Dic163C22⋊Q812yes—
166C2×C163—42no—
167C2×C163C2×C22⋊Q811yescharacteristic
168C2×C163#57611yescharacteristic
169C2×C163#57711yescharacteristic
170D163—22no—
171D163—42no—
172C163#308111yescharacteristic, complemented, Sylow
173#3081—1631nomaximal, Sylow
174#544—1631no—
175#518—1631no—
176#478—1631no—
177#478—1631no—
178#478—1631no—
179#478D16311yescharacteristic, complemented
180#476—1631no—
181#476—1631no—
182#474—1631no—
183#474—1631no—
184#428—1631no—
185#427—1631no—
186#424—1631no—
187#418—1631no—
188C2×#102—1631no—
189C22×D4—1631no—
190C2×M16—1631no—
191C22×C8—1631no—
192C4⋊Q8—3261no—
193C4⋊Q8—1631no—
194C4⋊Q8—1631no—
195#125—3261no—
196#125—1631no—
197#125C2×D16311yescharacteristic, complemented
198C22⋊Q8—3261no—
199C22⋊Q8—3261no—
200C4×D4—3261no—
201C4×D4—3261no—
202C4×D4—1631no—
203C4×D4—1631no—
204C4×D4—1631no—
205C4×D4C2×D16311yescharacteristic
206#117—1631no—
207#117—1631no—
208#117—1631no—
209C2×C4⋊C4—1631no—
210C2×C22⋊C4—3261no—
211C2×C22⋊C4—3261no—
212#107—1631no—
213#107—1631no—
214#107—1631no—
215#107C2×D16311yescharacteristic, complemented
216#106—1631no—
217#106—1631no—
218#106—1631no—
219#106C2×D16311yescharacteristic, complemented
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222C4⋊C8—1631no—
223C4⋊C8C2×D16311yescharacteristic, complemented
224#102—1631no—
225#102—1631no—
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102C2×D16311yescharacteristic
231#102C2×D16311yescharacteristic
232C24—3261no—
233C2×Q8—3261no—
234C2×Q8—3261no—
235C2×D4—3262no—
236C2×D4—3261no—
237C2×D4—1634no—
238C2×D4—1631no—
239C2×D4C22×D16311yescharacteristic
240C22×C4—21no—
241C22×C4—3261no—
242C22×C4—3261no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—1631no—
246M16—3262no—
247C2×C8—3262no—
248C2×C8—1631no—
249C2×C8—1631no—
250C2×C8C22×D16311yescharacteristic
251C2×C8C22×D16311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—1631no—
260C4⋊C4—1631no—
261C4⋊C4—1631no—
262C4⋊C4—1631no—
263C4⋊C4—1631no—
264C4⋊C4C22×D16311yescharacteristic
265C4⋊C4C22×D16311yescharacteristic
266C4⋊C4C22×D16311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—1631no—
274C22⋊C4—1631no—
275C42—3261no—
276C42—3261no—
277C42—1631no—
278C42C22×D16311yescharacteristic
279C23—21no—
280C23—13041no—
281C23—13041no—
282C23—3264no—
283C23—3261no—
284C23—1631no—
285Q8—13041no—
286Q8—13041no—
287D4—21no—
288D4—3264no—
289D4—3261no—
290D4—1632no—
291D4Q8×D16312yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—13041no—
297C2×C4—13041no—
298C2×C4—13041no—
299C2×C4—13041no—
300C2×C4—3261no—
301C2×C4—3261no—
302C2×C4—3261no—
303C2×C4—1631no—
304C2×C4—1631no—
305C2×C4—1631no—
306C2×C4—1632no—
307C2×C4—1632no—
308C2×C4D4×D16311yescharacteristic
309C2×C4D4×D16311yescharacteristic
310C2×C4C23×D16311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3261no—
314C8—3261no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—13044no—
319C22—3261no—
320C22—3261no—
321C22—3262no—
322C22—6521no—
323C22—6521no—
324C22—1632no—
325C22C2×D4×D16311yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—3261no—
332C4—3261no—
333C4—6521no—
334C4—6521no—
335C4—6521no—
336C4—1632no—
337C4C2×Q8×D16311yescharacteristic
338C4#41×D16311yescharacteristic
339C2—42no—
340C2—3262no—
341C2—6522no—
342C2C22⋊Q8×D16311yescharacteristic
343C2#2378246911yescharacteristic
344C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×D163C22
hypercenter#478D163
hypocenterC163#3081
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23782295C2semidirect
3. #23781967C2semidirect
4. #23781964C2semidirect
5. #23781962C2semidirect
6. #23781956C2semidirect
7. #23781953C2semidirect
8. #23781945C2semidirect
9. #23781944C2semidirect
10. #23781706C2semidirect
11. #23781700C2semidirect
12. #23781690C2semidirect
15. #478×C163C2semidirect
19. #19104735C22semidirect
21. #19104734C22semidirect
22. #19104726C22semidirect
37. #19104668C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. #19104667C22semidirect
42. #19104666C22semidirect
43. C4⋊(C163⋊C8)C22semidirect
44. Dic163⋊C8C22semidirect
45. Dic163⋊C8C22semidirect
52. #125×C163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. C4⋊C8×C163C22semidirect
65. D4×D163Q8semidirect
172. C163#3081semidirect
179. #478D163semidirect
197. #125C2×D163semidirect
215. #107C2×D163semidirect
219. #106C2×D163semidirect
223. C4⋊C8C2×D163semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237822952
#237821372
#237819672
#237819642
#237819622
#237819562
#237819532
#237819452
#237819442
#237817062
#237817002
#237816902
#237816682
#102×D1632
#478×C1632
#3081163